Partitions of Baire space into compact sets

Under $\text{CH}$ we construct a partition of Baire space into compact sets, which is indestructible by countably supported iteration and product of Sacks forcing of any length, answering a question of Newelski. Further, we present an in-depth isomorphism-of-names argument for $\text{spec}({\mathfrak{a}}_{\text{T}})=\{{\mathrm{\aleph }}_1,\mathfrak{c}\}$ in the product-Sacks model. Finally, we prove that Shelah’s ultrapower model for the consistency of $\mathfrak{d}<\mathfrak{a}$ also satisfies $\mathfrak{a}={\mathfrak{a}}_{\text{T}}$. Thus, consistently ${\mathrm{\aleph }}_1<\mathfrak{d}<\mathfrak{a}={\mathfrak{a}}_{\text{T}}$ holds relative to a measurable.